Inverse problems for quasi-variational inequalities

In this short note, our aim is to investigate the inverse problem of parameter identification in quasi-variational inequalities. We develop an abstract nonsmooth regularization approach that subsumes the total variation regularization and permits the identification of discontinuous parameters. We study the inverse problem in an optimization setting using the output-least squares formulation. We prove the existence of a global minimizer and give convergence results for the considered optimization problem. We also discretize the identification problem for quasi-variational inequalities and provide the convergence analysis for the discrete problem. We give an application to the gradient obstacle problem.     

Representation of attraction elements in abstract attainability problems with asymptotic constraints

We consider an abstract attainability problem with asymptotic constraints in a topological space. We construct an extension in the class of ultrafilters of widely interpreted measurable spaces. We study an example of a static problem on the asymptotic attainability in the class of layer functions.     

Path regularity and explicit convergence rate for BSDE with truncated quadratic growth

We consider backward stochastic differential equations with drivers of quadratic growth (qgBSDE). We prove several statements concerning path regularity and stochastic smoothness of the solution processes of the qgBSDE, in particular we prove an extension of Zhang’s path regularity theorem to the quadratic growth setting. We give explicit convergence rates for the difference between the solution of a qgBSDE and its truncation, filling an important gap in numerics for qgBSDE. We give an alternative proof of second order Malliavin differentiability for BSDE with drivers that are Lipschitz continuous (and differentiable), and then derive an analogous result for qgBSDE.     

On superalgebras over operads

We show that the class of varieties of multioperator superalgebras, defined by multilinear identities and considered up to rational equivalence, coincides with the class of varieties of superalgebras over an operad. We define the concepts of Grassmann and Clifford envelopes in the most general case and study their properties. We define the concept of module over a superalgebra over an operad and the concept of universal enveloping superalgebra for an algebra over an operad and study their properties.     

New concepts in neutrosophic graphs with application

A neutrosophic set is a generalization of an intuitionistic fuzzy set. Neutrosophic models give more flexibility, precisions and compatibility to the system as compared to intuitionistic fuzzy models. In this research study, we apply the concept of neutrosophic sets to graphs and discuss certain concepts of single-valued neutrosophic graphs. We illustrate the concepts by several examples. We investigate some interesting properties. We describe an application of single-valued neutrosophic graph in decision making process. We also present the procedure of our proposed method as an algorithm.     

On the distribution tail of an integrated risk model: A numerical approach

We consider an insurance risk process with the possibility to invest the capital reserve into a portfolio consisting of a risky asset and a riskless asset. The stock price is modelled by an exponential Lévy process and the riskless interest rate is assumed to be constant. We aim at the risk assessment of the integrated risk process in terms of a high quantile or the far out distribution tail. We indicate an application to an optimal investment strategy of an insurer.     

Using a Conic Formulation for Finding Steiner Minimal Trees

We present a new mathematical programming formulation for the Steiner minimal tree problem. We relax the integrality constraints on this formulation and transform the resulting problem (which is convex, but not everywhere differentiable) into a standard convex programming problem in conic form. We consider an efficient computation of an ε-optimal solution for this latter problem using an interior-point algorithm.     

Estimates on the Distribution of the Condition Number of Singular Matrices

We exhibit some new techniques to study volumes of tubes about algebraic varieties in complex projective spaces. We prove the existence of relations between volumes and Intersection Theory in the presence of singularities. In particular, we can exhibit an average Bezout Equality for equidimensional varieties. We also state an upper bound for the volume of a tube about a projective variety. As a main outcome, we prove an upper bound estimate for the volume of the intersection of a tube with an equidimensional projective algebraic variety. We apply these techniques to exhibit upper bounds for the probability distribution of the generalized condition number of singular complex matrices.     

Liquidity Costs: A New Numerical Methodology and an Empirical Study

We consider rate swaps which pay a fixed rate against a floating rate in the presence of bid-ask spread costs. Even for simple models of bid-ask spread costs, there is no explicit strategy optimizing an expected function of the hedging error. We here propose an efficient algorithm based on the stochastic gradient method to compute an approximate optimal strategy without solving a stochastic control problem. We validate our algorithm by numerical experiments. We also develop several variants of the algorithm and discuss their performances in terms of the numerical parameters and the liquidity cost.     

Automorphism groups of polycyclic-by-finite groups and arithmetic groups

We show that the outer automorphism group of a polycyclic-by-finite group is an arithmetic group. This result follows from a detailed structural analysis of the automorphism groups of such groups. We use an extended version of the theory of the algebraic hull functor initiated by Mostow. We thus make applicable refined methods from the theory of algebraic and arithmetic groups. We also construct examples of polycyclic-by-finite groups which have an automorphism group which does not contain an arithmetic group of finite index. Finally we discuss applications of our results to the groups of homotopy self-equivalences of K(Γ,1)-spaces and obtain an extension of arithmeticity results of Sullivan in rational homotopy theory.     

A multidomain discretization method with local mesh refinement

We introduce a nonconforming finite element formulation of asecond-order elliptic problem in the plane. The problem is posedon a domain which is partitioned into disjoint subregions andthe discrete problem is based on an independent triangulationby subregion. We use an integral matching condition at the interfaces.We prove quasi-optimal error estimates and propose an iterativesubstructuring algorithm to solve the system of linear equationsrelated to our discrete problem. The algorithm and its convergenceanalysis rely on an abstract framework of the additive Schwarzmethod.     

Risk-sensitive control of continuous time Markov chains

We study risk-sensitive control of continuous time Markov chains taking values in discrete state space. We study both finite and infinite horizon problems. In the finite horizon problem we characterize the value function via Hamilton Jacobi Bellman equation and obtain an optimal Markov control. We do the same for infinite horizon discounted cost case. In the infinite horizon average cost case we establish the existence of an optimal stationary control under certain Lyapunov condition. We also develop a policy iteration algorithm for finding an optimal control.     

Numerical Optimal Harvesting for a Periodic Age-Structured Population Dynamics with Logistic Term

We investigate an optimal harvesting problem for age-structured population dynamics with logistic term and periodic vital rates. We use first-order necessary optimality conditions in order to derive an algorithm to approximate the optimal harvesting effort. We present corresponding numerical experiments.     

Outerplanar Thrackles

We show that a graph drawing is an outerplanar thrackle if and only if, up to an inversion in the plane, it is Reidemeister equivalent to an odd musquash. This establishes Conway’s thrackle conjecture for outerplanar thrackles. We also extend this result in two directions. First, we show that no pair of vertices of an outerplanar thrackle can be joined by an edge in such a way that the resulting graph drawing is a thrackle. Secondly, we introduce the notion of crossing rank; drawings with crossing rank 0 are generalizations of outerplanar drawings. We show that all thrackles of crossing rank 0 are outerplanar. We also introduce the notion of an alternating cycle drawing, and we show that a thrackled cycle is alternating if and only if it is outerplanar.     

Effect of Delayed Payments (Trade Credit) on Order Quantities

We address the problem of determining the economic order quantity when the vendor permits delay in payment. Some early researchers argued that the best order quantity is invariant with respect to the trade credit. Others argued that the order quantity should increase as the delay in payment increases. We analyse this problem using the discounted cash-flow approach, and provide clarification on the inconsistencies between these approaches. First, we show that the best order quantity is an increasing function of the permitted delay in payment. We also show that an approach suggested by Chand and Ward not only yields an upper bound on the optimum, but also provides robust results. Though the classical square-root formula disregards trade-credit information, under some circumstances, it will yield better results than the formulation taking into account that information. We illustrate this anomaly with an example, and provide analytical explanation for it. We also discuss some potential conceptual pitfalls in using the average cost analysis as an approximation to the discounted cash-flow approach.     

Fractal methods and the problem of estimating scaling exponents: A new approach based on upper and lower linear bounds

The characterization of irregular objects with fractal methods often leads to the estimation of the slope of a function which is plotted versus a scale parameter. The slope is usually obtained with a linear regression. The problem is that the fit is usually not acceptable from the statistical standpoint. We propose a new approach in which we use two straight lines to bound the data from above and from below. We call these lines the upper and lower linear bounds. We propose to define these bounds as the solution of an optimization problem. We discuss the solution of this problem and we give an algorithm to obtain its solution. We use the difference between the upper and lower linear bounds to define a measure of the degree of linearity in the scaling range. We illustrate our method by analyzing the fluctuations of the variogram in a microresistivity well log from an oil reservoir in the North Sea.     

Formula for the number of eigenvalues of a three-particle Schrödinger operator on a lattice

We consider a system of three arbitrary quantum particles on a three-dimensional lattice that interact via short-range attractive potentials. We obtain a formula for the number of eigenvalues in an arbitrary interval outside the essential spectrum of the three-particle discrete Schrödinger operator and find a sufficient condition for the discrete spectrum to be finite. We give an example of an application of our results.     

On integrability of a heavy rigid body sinking in an ideal fluid

We consider a rigid body possessing 3 mutually perpendicular planes of symmetry, sinking in an ideal fluid. We prove that the general solution to the equations of motion branches in the complex time plane, and that the equations consequently are not algebraically integrable. We show that there are solutions with an infinitely-sheeted Riemannian surface.     

遍历性及其应用──(Ⅰ)基本性质(英)

在一般意义下，给出了函数的遍历性定义．指出遍历函数是比概周期函数、渐近概周期函数及弱概周期函数更广的一类函数．文章讨论了遍历函数的一些性质，其中一个主要结果是给出了一个不等式的明确表达式．     

Equations with Random Gaussian Operators with Unbounded Mean Value

We consider an equation in a Hilbert space with a random operator represented as a sum of a deterministic, closed, densely defined operator and a Gaussian strong random operator. We represent a solution of an equation with random right-hand side in terms of stochastic derivatives of solutions of an equation with deterministic right-hand side. We consider applications of this representation to the anticipating Cauchy problem for a stochastic partial differential equation.